The present invention generally relates to a method of simulating the electric characteristics of a semiconductor device by a computer, and more particularly, to a method of simulating an impact ionization phenomenon of carriers in a semiconductor device.
A computer simulation method of simulating the electric characteristics in a semiconductor device is described in Ryo Dan ed., "Process Device Simulation Technique", pp. 91-134 (hereunder referred to as a first literature). In this example, a region to be analyzed is divided into a mesh of subsections or elements. Further, at each mesh point (or node), a Poisson equation, an electron current continuity equation, and a hole current continuity equation are discretized. Moreover, these equations are linearized and then solved by using a Newton's method.
What, is called a control volume method described on page 114 of the first literature is widely used for the discretization of equations.
An approach for introducing a term concerning carrier generation by impact ionization under a high electric field into the current continuity equations discretized by such a control-volume method is explained in, for example, "Numerical Formulation" section on page 2077 of S. E. Laux and B. M. Grossman, "A General Control-Volume Formulation for Modeling Impact Ionization in Semiconductor Transport," IEEE Trans. Electron Devices, vol. ED-32, no. 10, pp. 2076-2082 (hereunder referred to as a second literature).
The current density of electric current flowing through a mesh edge (or side or branch) is given by an equation (3.68) described in the first literature and an equation (3) described on page 2077 in the second literature by the use of a Scharfetter-Gummel scheme described on pages 119-122 of the first literature.
A simulation of an impact ionization phenomenon occurring in a semiconductor device in a stationary state can be realized by employing these equations and a current continuity equation in a case where impact ionization components are main components in the stationary state, and the Poisson equation as simultaneous equations, and then solving the simultaneous equations by using a suitable boundary condition.
However, the aforementioned prior art method of simulating an impact ionization phenomenon in a semiconductor device has a problem in that instability occurs when calculating data representing a state where a noticeable impact ionization is caused in a high electric field.
The prior art methods disclosed in the first and the second literature references employ a formulation by which an amount of generated carriers is first evaluated by using the current density of electric currents flowing from control volumes, and by which the evaluated amounts of generated carriers are then assigned or partitioned to the control volumes that act as sources of electric current, and a positive feedback is locally performed. Thus, when the ionization coefficient corresponding to the electron density at a mesh point increases due to a rise in the magnitude of the electric field, namely, when the electric field rises so that the ionization coefficient increases, the simulation comes upon a situation in which the coefficient corresponding to the carrier density in the discretized current continuity equation becomes zero. The occurrence of this situation means that the carrier density can have an arbitrary value. As a result, the simulation becomes unstable.
As a measure to avoid an occurrence of the instability, there has been devised a prior art method of shifting a point, at which the coefficient corresponding to the carrier density becomes zero, toward a high electric field side by decreasing the distance L between mesh points. However, generally, the ionization coefficient .alpha. increases exponentially with an increase in the magnitude of the electric field E. Thus, in the case of employing this method, the required number of elements of a mesh increases exponentially with an increase in the magnitude of the electric field E. Consequently, a computation time duration exponentially increases. Therefore, this prior art method is of no practical use.